Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2009 Grade 5 Number Sense
Problem
Solution
1) A rat runs in a 2 meter tube that has a number line below it marking off the lengths as shown on the number line.
(Use the above number line)
A student takes data of the rat's position every few seconds. She writes the following.
  • The rat starts at 1/4 meter.
  • Add 0.95m to get to its next point.
  • To find its next point on the number line take this last value and multiply it by 150%.
  • From this point subtract the sum of 1/3 + 1/6 meter.
Where is the rat now located on the number line? 		1. Rat starts at .25 meter
2. .25 + .95 = 1.20 meter
3. 1.5 X 1.2 = 1.8 meter
4. 1/3 + 1/6 meter and convert to a decimal = .5 meter
5. Rat is at 1.8 - .5 = 1.3 meter

2) A bank teller doesn't realize that on his computer screen the digit in the one's place is not printing. So he reads $12.00 when the actual number could be $126.00. He doesn't see the 6. A customer makes him aware of the problem when she doesn't get the right amount of cash. What's the most the teller could owe his customer if the number he saw on his screen looked like more than $10 but less than $100? 		1. The maximum 2-digit number is 99.
2. Add 9 to make 999.
3. Amount owed = 999 - 99 = $900.

Problem
Solution
3) PINs or personal identification numbers are needed everywhere once you start setting up accounts for just about anything. Eric decides he needs a plan for the PINs he will need. His first PIN will be 7862 for reasons known only to him. After that every new PIN will be obtained from 7862 by adding the next prime number , for example 7864 is his second pin (he added 2). All he needs to remember is where he is in the list of prime numbers, and he knows all his PINs. Where in the list is the first PIN that doesn't have an 8 in it? To not have an "8" in the PIN means the prime number and 62 must be more than 100, because the second digit of his original PIN has an 8.
The prime number must be greater than 100 - 62 = 38.
The first 13 prime numbers are:
    2 3 5 7 11 13 17 19 23 29 31 37 41
The first prime number greater than 38 is 41.
    It is 13th on the list.

4) A small country prints only two kinds of stamps, 7 cent stamps and 13 cent stamps. One letter requires 64 cents postage. How many of each stamp should you ask for so that you can mail the letter and spend the least amount of money? 		Filling in the table, we have:
# 7-cent
stamps		# 13-cent
stamps		  Total  
10070
8169
6268
5374
4470
0565

You should use 5 13-cent stamps.
5) Sonja swims in races. This season she notices that she drops in time for her butterfly at each race by 1/3 of the time she dropped at the previous race. She began the season at 1 minute and 25 seconds, the second race she did 1 minute and 7 seconds, an 18 second drop. What should her time be in the 5th race? Express you answer to the nearest one hundredth of a second. Method 1: Compute the time for each race
  1. Race 1 her time was 85 seconds
  2. Race 2 she dropped by 18 seconds for a time of 67 seconds
  3. Race 3 she dropped by 6 seconds for a time of 61 seconds
  4. Race 4 she dropped by 2 seconds for a time of 59 seconds
  5. Race 5 she dropped by 2/3 second for a time of 58.33 seconds.
Method 2: Sum the seconds lost and subtract from her first race time:
  1. Race 1 time was 85 seconds
  2. The total of the amounts she dropped was 18 + 6 + 2 + 2⁄3 seconds = 262⁄3 seconds
  3. 85 - 262⁄3 = 58.33 seconds